A regime of very long meandering positive and negative streamwise velocity fluctuations, that we term ‘superstructures’, are found to exist in the log and lower wake regions of turbulent boundary layers. Measurements are made with a spanwise rake of 10 hot-wires in two separate facilities (spanning more than a decade of Reτ) and are compared with existing PIV and DNS results. In all cases, we note evidence of a large-scale stripiness in the streamwise velocity fluctuations. The length of these regions can commonly exceed 20δ. Similar length scales have been previously reported for pipes and DNS channel flows. It is suggested that the true length of these features is masked from single-point statistics (such as autocorrelations and spectra) by a spanwise meandering tendency. Support for this conjecture is offered through the study of a synthetic flow composed only of sinusoidally meandering elongated low- and high-speed regions. From detailed maps of one-dimensional spectra, it is found that the contribution to the streamwise turbulence intensities associated with the superstructures appears to be increasingly significant with Reynolds number, and scales with outer length variables (δ). Importantly, the superstructure maintains a presence or footprint in the near-wall region, seeming to modulate or influence the near-wall cycle. This input of low-wavenumber outer-scaled energy into the near-wall region is consistent with the rise in near-wall streamwise intensities, when scaled with inner variables, that has been noted to occur with increasing Reynolds number. In an attempt to investigate these structures at very high Reynolds numbers, we also report on recent large-scale sonic anemometer rake measurements, made in the neutrally stable atmospheric surface layer. Preliminary results indicate that the superstructure is present in the log region of this atmospheric flow at Reτ = 6.6×105, and has a size consistent with outer scaling.

When a solid plate, with a boundary condition of no normal flow through it, is introduced parallel to a shear layer it is normally expected to exert a stabilizing influence on any inviscid linearly unstable waves. In this paper we present an example of an absolutely unstable boundary-layer flow that can be made more absolutely unstable by the addition of a plate parallel to the original flow and far from the boundary layer itself. In particular, the addition of the plate is found to increase the growth rate of the absolute instability of the original boundary-layer flow by an order of magnitude for long waves. This phenomenon is illustrated using piecewise-linear inviscid basic-flow profiles, for which analytical dispersion relations have been derived. Long-wave stability theories have been developed in several limits clarifying the mechanisms underlying the behaviour and establishing its generic nature. The class of flows expected to exhibit this phenomenon includes a class found recently to have an exponential growth of disturbances in the wall-normal direction, owing to the approach of certain saddle-points to certain branch-cuts in the complex-wavenumber plane. The theory also suggests that a convectively unstable flow in an infinite domain can be converted, in some circumstances, into an absolutely unstable flow when the domain is made finite by the addition of a plate, however far away the plate is.

The relaxation of a dewetting contact line is investigated theoretically in the so-called ‘Landau–Levich’ geometry in which a vertical solid plate is withdrawn from a bath of partially wetting liquid. The study is performed in the framework of lubrication theory, in which the hydrodynamics is resolved at all length scales (from molecular to macroscopic). We investigate the bifurcation diagram for unperturbed contact lines, which turns out to be more complex than expected from simplified ‘quasi-static’ theories based upon an apparent contact angle. Linear stability analysis reveals that below the critical capillary number of entrainment, Cac, the contact line is linearly stable at all wavenumbers. Away from the critical point, the dispersion relation has an asymptotic behaviour σ∝|q| and compares well to a quasi-static approach. Approaching Cac, however, a different mechanism takes over and the dispersion evolves from ∼|q| to the more common ∼q2. These findings imply that contact lines cannot be described using a universal relation between speed and apparent contact angle, but viscous effects have to be treated explicitly.

The principal objective of this paper is to study some unsteady characteristics of an interaction between an incident oblique shock wave impinging on a laminar boundary layer developing on a flat plate. More precisely, this paper shows that some unsteadiness, in particular the low-frequency unsteadiness, originates in a supercritical Hopf bifurcation related to the dynamics of the separated boundary layer. Various direct numerical simulations were carried out of a shock-wave/laminar-boundary-layer interaction (SWBLI). Three-dimensional unsteady Navier–Stokes equations are numerically solved with an implicit dual time stepping for the temporal algorithm and high-order AUSMPW+ scheme for the spatial discretization. A parametric study on the oblique shock-wave angle has been performed to characterize the unsteady behaviour onset. These numerical simulations have shown that starting from the incident shock angle and the spanwise extension, the flow becomes three-dimensional and unsteady. A linearized global stability analysis is carried out in order to specify and to find some characteristics observed in the direct numerical simulation. This stability analysis permits us to show that the physical origin generating the three-dimensional characters of the flow results from the existence of a three-dimensional stationary global instability.

Hazardous natural flows such as snow-slab avalanches, debris flows, pyroclastic flows and lahars are part of a much wider class of dense gravity-driven granular free-surface flows that frequently occur in industrial processes as well as in foodstuffs in our kitchens! This paper investigates the formation of oblique granular shocks, when the oncoming flow is deflected by a wall or obstacle in such a way as to cause a rapid change in the flow height and velocity. The theory for non-accelerative slopes is qualitatively similar to that of gasdynamics. For a given deflection angle there are three possibilities: a weak shock may form close to the wall; a strong shock may extend across the chute; or the shock may detach from the tip. Weak shocks have been observed in both dense granular free-surface flows and granular gases. This paper shows how strong shocks can be triggered in chute experiments by careful control of the downstream boundary conditions. The resulting downstream flow height is much thicker than that of weak shocks and there is a marked decrease in the downstream velocity. Strong shocks therefore dissipate much more energy than weak shocks. An exact solution for the angle at which the flow detaches from the wedge is derived and this is shown to be in excellent agreement with experiment. It therefore provides a very useful criterion for determining whether the flow will detach. In experimental, industrial and geophysical flows the avalanche is usually accelerated, or decelerated, by the net effect of the gravitational acceleration and basal sliding friction as the slope inclination angle changes. The presence of these source terms necessarily leads to gradual changes in the flow height and velocity away from the shocks, and this in turn modifies the local Froude number of the flow. A shock-capturing non-oscillating central method is used to compute numerical solutions to the full problem. This shows that the experiments can be matched very closely when the source terms are included and explains the deviations away from the classical oblique-shock theory. We show that weak shocks bend towards the wedge on accelerative slopes and away from it on decelerative slopes. In both cases the presence of the source terms leads to a gradual increase in the downstream flow thickness along the wedge, which suggests that defensive dams should increase in height further down the slope, contrary to current design criteria but in accordance with field observations of snow-avalanche deposits from a defensive dam in Northwestern Iceland. Movies are available with the online version of the paper.

Thiria, Goujon-Durand & Wesfreid (J. Fluid Mech. vol. 560, 2006, p. 123), it was shown that vortex shedding from a rotationally oscillating cylinder at moderate Reynolds number can be characterized by the spatial coexistence of two distinct patterns, one of which is related to the forcing frequency in the near wake and the other to a frequency close to the natural one for the unforced case downstream of this locked region. The existence and the modification of these wake characteristics were found to be strongly affected by the frequency and the amplitude of the cylinder oscillation. In this paper, a linear stability analysis of these forced regimes is performed, and shows that the stability characteristics of such flows are governed by a strong mean flow correction which is a function of the oscillation parameters. We also present experiments on the spatial properties of the global mode and on the selection of the vortex shedding frequency as a function of the forcing conditions for Re = 150. Finally, we elucidate a diagram of locked and non-locked states, for a large range of frequencies and amplitudes of the oscillation.

The principal nonlinear correction to the dynamics of gravity waves on an irrotational fluid is traditionally derived as a non-resonant perturbation solution to the Stokes expansion. When the problem is reformulated in the Hamiltonian description and limited to moderately collimated random waves over infinite depth, the perturbation term assumes a very simple and descriptive form. The sum-frequency component for the surface height is just a bilinear product of the height with the associated scalar strain, and the accompanying term in the potential is half the time derivative of the squared linear height. This solution is exact in one surface dimension and remains quite accurate for long-crested waves in two dimensions, with an error small to second order in the angular spread of constituent wave vectors. It is a natural generalization for random, disordered wave ensembles of the second-order Stokes solution, and its effect is to sharpen the random crests and to flatten the troughs. For wave sets of narrow relative bandwidth the difference-frequency component consists of a negligible elevation term and a non-negligible potential term whose gradient is the surface value of the volume return flow balancing the quadratic wave transport of fluid.

Electric conduction from an electrolyte solution into a charge selective solid, such as ion exchange membrane or electrode, becomes unstable when the electrolyte concentration near the interface approaches zero owing to diffusion limitation. The sequence of events leading to instability is as follows: upon the decrease of the interface concentration, the electric double layer at the interface transforms from its common quasi-equilibrium structure to a different, non-equilibrium one. The key feature of this new structure is an extended space charge added to the usual one of the quasi-equilibrium electric double layer. The non-equilibrium electro-osmotic slip related to this extended space charge renders the quiescent conductance unstable. A unified asymptotic picture of the electric double-layer undercurrent, encompassing all regimes from quasi-equilibrium to the extreme non-equilibrium one, is developed and employed for derivation of a universal electro-osmotic slip formula. This formula is used for a linear stability study of quiescent electric conduction, yielding the precise parameter range of instability, compared with that in the full electroconvective formulation. The physical mechanism of instability is traced both kinematically, in terms of non-equilibrium electro-osmotic slip, and dynamically, in terms of forces acting in the electric double layer.

Direct numerical simulation (DNS) is used to investigate turbulent Taylor–Couette (TC) flow. A simulation was run for a Reynolds number of 3200 in an apparatus with a radius ratio of η = 0.617 and an aspect ratio of 4.58, which assumed a vortex pair wavelength of 2.29. Results reported include the mean velocity, velocity fluctuation intensities, Reynolds stress budgets, and visualizations of the instantaneous velocity fluctuation field. Secondary near-wall vortex pairs are observed near to the cylinder in addition to the Taylor vortex (TV) motion. Weaker evidence of secondary vortices is found at the outer cylinder where a banded structure has been identified. The azimuthal wall shear stress component shows large peaks and valleys at stagnation points on the surface of both cylinders where flow from neighbouring vortices impacts on the respective wall. These stagnation points correspond to locations where the secondary vortices have been identified. The effect of the mean TV motion is reflected in the Reynolds stress budgets which are similar to but more complex than those of two-dimensional boundary layers. Visualization of the turbulent velocity fluctuations reveals near-wall streaks at the inner cylinder.

We consider the harmonic and subharmonic modulated surface waves that appear upon horizontal vibration along the surface of the liquid in a two-dimensional large-aspect-ratio (length large compared to depth) container, whose depth is large compared to the wavelength of the surface waves. The analysis requires us also to consider an oscillatory bulk flow and a viscous mean flow. A weakly nonlinear description of the harmonic waves is made which provides the threshold forcing amplitude to trigger harmonic instabilities, which are of various qualitatively different kinds. A linear analysis provides the threshold amplitude for the appearance of subharmonic waves through a subharmonic instability. The results obtained are used to make several specific qualitative and quantitative predictions.

Two-dimensional global eigenmodes are used as a projection basis both for analysing the dynamics and building a reduced model for control in a prototype separated boundary-layer flow. In the present configuration, a high-aspect-ratio smooth cavity-like geometry confines the separation bubble. Optimal growth analysis using the reduced basis shows that the sum of the highly non-normal global eigenmodes is able to describe a localized disturbance. Subject to this worst-case initial condition, a large transient growth associated with the development of a wavepacket along the shear layer followed by a global cycle related to the two unstable global eigenmodes is found. The flow simulation procedure is coupled to a measurement feedback controller, which senses the wall shear stress at the downstream lip of the cavity and actuates at the upstream lip. A reduced model for the control optimization is obtained by a projection on the least stable global eigenmodes, and the resulting linear-quadratic-Gaussian controller is applied to the Navier–Stokes time integration. It is shown that the controller is able to damp out the global oscillations.

Development, experimental implementation, and the results of reduced-order model based feedback control of subsonic shallow cavity flows are presented and discussed. Particle image velocimetry (PIV) data and the proper orthogonal decomposition (POD) technique are used to extract the most energetic flow features or POD eigenmodes. The Galerkin projection of the Navier–Stokes equations onto these modes is used to derive a set of nonlinear ordinary differential equations, which govern the time evolution of the eigenmodes, for the controller design. Stochastic estimation is used to correlate surface pressure data with flow-field data and dynamic surface pressure measurements are used to estimate the state of the flow. Five sets of PIV snapshots of a Mach 0.3 cavity flow with a Reynolds number of 105 based on the cavity depth are used to derive five different reduced-order models for the controller design. One model uses only the snapshots from the baseline (unforced) flow while the other four models each use snapshots from the baseline flow combined with snapshots from an open-loop sinusoidal forcing case. Linear-quadratic optimal controllers based on these models are designed to reduce cavity flow resonance and are evaluated experimentally. The results obtained with feedback control show a significant attenuation of the resonant tone and a redistribution of the energy into other modes with smaller energy levels in both the flow and surface pressure spectra. This constitutes a significant improvement in comparison with the results obtained using open-loop forcing. These results affirm that reduced-order model based feedback control represents a formidable alternative to open-loop strategies in cavity flow control problems even in its current state of infancy.

The nonlinear Navier–Stokes equations governing steady, laminar, axisymmetric flow past a deformable bubble are solved by the Galerkin finite-element method simultaneously with a set of elliptic partial differential equations governing boundary-fitted mesh. For Reynolds number 20 ≤ Re ≤ 500, numerical solutions of spherical-cap bubbles are obtained at capillary number Ca = 1. Increasing Ca to 2 leads to a highly curved, cusp-like bubble rim that seems to correspond to skirt formation. The computed steady, axisymmetric spherical-cap bubbles with closed, laminar wakes compare reasonably with the available experimental results, especially for Re ≤ 100. By exploring the parameter space (for Re ≤ 200), a sufficient condition for steady axisymmetric solutions of bubbles with the spherical-cap shape is found to be roughly Ca > 0.4. The basic characteristics of spherical-cap bubbles of Ca ≥ 0.5, for a given Re ≥ 50, are found to be almost independent of the value of Ca (or Weber number We ≡ Re Ca). At a fixed Re ≥ 50, continuation by increasing Ca (or We) from a spherical bubble solution cannot lead to solutions of spherical-cap bubbles, but rather to a turning point at We slightly greater than 10 where the solution branch folds back to reduced values of Ca (or We). Yet continuation by reducing Ca (or We) from a spherical-cap bubble solution cannot arrive at a spherical bubble solution for Re ≥ 50, but rather at solutions with bubbles having more complicated shapes such as a sombrero, etc. Without thorough examinations of the solution stability, multiple steady axisymmetric solutions are shown to exist in the parameter space for a given set of parameters.

We consider the propagation of a gravity current of density ρc from a lock of length x0 and height h0 into an ambient fluid of density ρa in a channel of height H. When the Reynolds number is large, the resulting flow is governed by the parameters ρc/ρa and H* = H/h0. We show that the shallow-water one-layer model, combined with a Benjamin-type front condition, provides a versatile formulation for the thickness and speed of the current, without any adjustable constants. The results cover in a continuous manner the range of light ρc/ρa ≪ 1, Boussinesq ρc/ρa ≈ 1, and heavy ρc/ρa ≫ 1 currents in a fairly wide range of depth ratio. We obtain analytical solutions for the initial dam-break or slumping stage of propagation with constant speed, and derive explicitly the trends for small and large values of the governing parameters. This reveals the main features: (a) the heavy current propagates faster and its front is thinner than for the light counterpart; (b) the speed of the heavy current depends little on H*, while that of the light current increases with H*; and (c) in the shallow ambient case (H* close to 1) the light current is choked to move with the thickness of half-channel, while the heavy current typically moves with an unrestricted smaller thickness. These qualitative predictions are in accord with previous observations, and some quantitative comparisons with available experimental and numerical simulations data also show fair agreement. However, given the paucity of the available data, the main deficiency of the model is the unknown practical limit of applicability. For large time, t, a self-similar propagation with t2/3 is feasible for both the heavy and light currents, but the thickness profiles display differences.

We report a theoretical investigation of the robustness of two-dimensional inviscid magnetohydrodynamic (MHD) flows at low magnetic Reynolds numbers with respect to three-dimensional perturbations. We use a combination of linear stability analysis and direct numerical simulations to analyse three problems, namely the flow in the interior of a triaxial ellipsoid, and two unbounded flows: a vortex with elliptical streamlines and a vortex sheet parallel to the magnetic field. The flow in a triaxial ellipsoid is found to present an exact analytical model which demonstrates both the existence of inviscid unstable three-dimensional modes and the stabilizing role of the magnetic field. The nonlinear evolution of the flow is characterized by intermittency typical of other MHD flows with long periods of nearly two-dimensional behaviour interrupted by violent three-dimensional transients triggered by the instability. We demonstrate, using the second model, that motion with elliptical streamlines perpendicular to the magnetic field becomes unstable with respect to the elliptical instability once the magnetic interaction parameter falls below a critical magnitude whose value tends to infinity as the eccentricity of the streamlines increases. Furthermore, the third model indicates that vortex sheets parallel to the magnetic field, which are unstable for any velocity and any magnetic field, emit eddies with vorticity perpendicular to the magnetic field. Whether the investigated instabilities persist in the presence of small but finite viscosity, in which case two-dimensional turbulence would represent a singular state of MHD flows, remains an open question.

The problem of linear wave scattering by an ice sheet of variable thickness floating on water of variable quiescent depth is considered by applying the Rayleigh–Ritz method in conjunction with a variational principle. By using a multi-mode expansion to approximate the velocity potential that represents the fluid motion, Porter & Porter (J. Fluid Mech. vol. 509, 2004, p. 145) is extended and the solution of the problem may be obtained to any desired accuracy. Explicit solution methods are formulated for waves that are obliquely incident on two-dimensional geometry, comparisons are made with existing work and a range of new examples that includes both total and partial ice-cover is considered.

We study numerically the effects of zero-mean stochastic gravity modulation on the mixing characteristics of two interdiffusing miscible Boussinesq fluids initially separated by a thin diffusion layer. The gravity modulation has a Gaussian probability distribution and is characterized by an exponentially damped cosine autocorrelation function, i.e. . The associated power spectrum is a Lorentzian with peak at ω and width λ. The flow is found to depend on the following parameters: the Grashof number, Gr, based on the viscous length scale, ; the Schmidt number, Sc; the correlation exponent, λ; and other geometric parameters. Even for extremely small Gr, we observe the propagation of gravity currents, Kelvin–Helmholtz (KH) and Rayleigh–Taylor (RT) instabilities. This is in contrast to the case of harmonic modulation considered in Part 1 (Siddavaram & Homsy J. Fluid Mech. vol. 562, 2006, p. 445) wherein these phenomena occur sequentially as Gr increases. The mixed volume is found to vary non-monotonically with the correlation exponent, λ, with narrow-band modulation having the largest mixed volume followed by harmonic modulation and then broadband modulation. This non-monotonicity of the mixed volume with λ is explained on the basis of the competition between the effects of excitation of lower frequencies, which lead to higher mixing, and the effects of the reduction in the energy content at the dominant frequency, which leads to reduced mixing. The value of the correlation coefficient, λ, at which the mixed volume is the largest is found to be independent of Gr. To understand the finer details of the mechanisms, we consider two- and three-frequency modulations. We find that increasing the amplitude of the secondary component when its frequency is smaller than that of the primary component leads to the occurrence of KH and RT instabilities at smaller Gr than that for the case of single-frequency modulation. We have understood the non-monotonic variation in the mixed volume by considering a three-frequency modulation, where one of the frequencies is smaller than the characteristic frequency and the other larger.

Using the Boussinesq approximation, a set of depth-integrated wave equations for long-wave propagation over a mud bed is derived. The wave motions above the mud bed are assumed to be irrotational and the mud bed is modelled as a highly viscous fluid. The pressure and velocity are required to be continuous across the water–mud interface. The resulting governing equations are differential–integral equations in terms of the depth-integrated horizontal velocity and the free-surface displacement. The effects of the mud bed appear in the continuity equation in the form of a time integral of weighted divergence of the depth-averaged velocity. Damping rates for periodic waves and solitary waves are calculated. For the solitary wave case, the velocity profiles in the water column and the mud bed at different phases are discussed. The effects of the viscous boundary layer above the mud–water interface are also examined.

An asymptotic analysis of Rayleigh–Taylor unstable ablation fronts encountered in inertial confinement fusion is performed in the case of a strong temperature dependence of the thermal conductivity. At leading order the nonlinear analysis leads to a free boundary problem which is an extension of the classical Rayleigh–Taylor instability with unity Atwood number and an additional potential flow of negligible density expelled perpendicular to the front. The nonlinear evolution of the front is analysed in two-dimensional geometry by a boundary integral method. The shape of the front develops a curvature singularity within a finite time, as for the Birkhoff–Rott equation for the Kelvin–Helmholtz instability.

This paper presents numerical results on the evolution of the solutions of the Fanno model for compressible pipe flow. The principal results concern the large-time behaviour when nonlinear effects are appreciable throughout the evolution. Our computations show that compression waves can be expected to evolve into travelling waves for large times whereas expansion waves cannot.